AI 中文总结
研究从射影直线到拉格朗日格拉斯曼流形的基于代数映射空间，通过箭图描述实现代数几何精细化，得到同构，其拓扑实现恢复经典同伦等价。
AI 中文摘要
我们给出了从射影直线\(\mathbb{P}^{1}\)到拉格朗日格拉斯曼流形（及其正交对应物）的基于代数映射空间的箭图描述。我们表明，我们的描述导致了实博特周期性中一些同伦等价的代数几何精细化。特别是，我们在拉尔森和瓦基尔的“朴素代数几何同伦范畴”中得到了一个同构，其拓扑实现（专门化为\(\mathbb{C}\)后）恢复了经典同伦等价\(\Omega^{2}(Sp/U) \simeq BO\times \mathbb{Z}\)和\(\Omega^{2}(O/U) \simeq BSp\times \mathbb{Z}\)。
英文摘要
We give a quiver description of the space of based algebraic maps from $\mathbb{P}^{1}$ to the Lagrangian Grassmannian (and its orthogonal counterpart). We show our descriptions lead to an algebro-geometric refinement of some of the homotopy equivalences in real Bott periodicity. In particular, we get an isomorphism in Larson and Vakil's ``naive algebro-geometric homotopy category'' whose topological realization (after specializing to $\mathbb{C}$) recovers the classical homotopy equivalences $Ω^{2}(Sp/U) \simeq BO\times \mathbb{Z}$ and $Ω^{2}(O/U) \simeq BSp\times \mathbb{Z}$.
